What are the Units for the five St. Venant Flow Terms in SWMM 5

What are the Units for the five St. Venant Flow Terms in SWMM 5

Understanding the Context: Dynamic Wave Routing

• St. Venant Equations: These are a set of partial differential equations that describe unsteady open-channel flow. They consist of the continuity equation (conservation of mass) and the momentum equation (conservation of momentum).
• Dynamic Wave Routing: This is the most comprehensive and accurate method for routing flows in SWMM 5 and InfoSWMM. It solves the full St. Venant equations, accounting for factors like backwater effects, flow reversals, and pressurized flow. It is the only routing method in InfoSWMM.
• Iterative Solution: Because the St. Venant equations are complex, SWMM 5 uses an iterative numerical method (e.g., Newton-Raphson) to solve them at each time step.

The Flow Equation and its Components

The core equation you presented is the heart of the iterative process:

(1) Q for the new iteration = (Q at the Old Time Step – DQ2 + DQ3 + DQ4 ) / ( 1.0 + DQ1 + DQ5)

This equation calculates the new flow (Q) in a link for the current iteration based on:

• Q at the Old Time Step: The flow calculated in the previous iteration.
• DQ1, DQ2, DQ3, DQ4, DQ5: These are the five St. Venant terms, representing different aspects of the momentum equation. They are calculated based on hydraulic conditions at the current iteration.

Breakdown of the St. Venant Terms and Their Units

As you correctly explained, the units are crucial for understanding the physical meaning of each term. Here's a more detailed breakdown:

• DQ2 (Pressure Term):

  • Equation: DQ2 = DT * GRAVITY * aWtd * (H2 – H1) / Length
  • Represents: The effect of the pressure gradient (difference in water surface elevation between the upstream and downstream nodes) on the flow.
  • Units:
    • DT (Time Step): seconds
    • GRAVITY: feet/second^2
    • aWtd (Weighted Area): feet^2 (This is a weighted average of the upstream and downstream cross-sectional areas)
    • (H2 – H1) (Head Difference): feet
    • Length: feet
    • Result: feet^3/second (CFS), which is a flow rate.

• DQ3 (Convective Acceleration Term):

  • Equation: DQ3 = 2 * Velocity * (aMid – aOld) * Sigma
  • Represents: The change in momentum due to the change in cross-sectional area along the link.
  • Units:
    • Velocity: feet/second
    • (aMid – aOld) (Change in Area): feet^2 (Difference between the area at the midpoint of the link at the current iteration and the area at the previous time step)
    • Sigma: unitless (This is a weighting factor, typically between 0.5 and 1.0, that accounts for the spatial variation of velocity and area along the link)
    • Result: feet^3/second (CFS), which is a flow rate.

• DQ4 (Local Acceleration Term):

  • Equation: DQ4 = DT * Velocity * Velocity * (aMid – aOld) * Sigma / Length
  • Represents: The change in momentum due to the change in velocity over time.
  • Units:
    • DT: seconds
    • Velocity: feet/second
    • Velocity: feet/second
    • (aMid – aOld): feet^2
    • Sigma: unitless
    • Length: feet
    • Result: feet^3/second (CFS), which is a flow rate.

• DQ1 (Friction Slope Term):

  • Equation: DQ1 = DT * GRAVITY * (n/PHI)^2 * Velocity / HydraulicRadius^1.333
  • Represents: The effect of friction on the flow, based on Manning's equation.
  • Units:
    • DT: seconds
    • GRAVITY: feet/second^2
    • (n/PHI)^2: second^2 / feet^(2/3) (n is Manning's roughness coefficient, PHI is a conversion factor related to the units used)
    • Velocity: feet/second
    • HydraulicRadius^1.333: feet^1.333
    • Result: Dimensionless (all units cancel out). This term acts as a scaling factor in the denominator of the main equation.

• DQ5 (Area Correction Term):

  • Equation: DQ5 = K * Q / Area / 2 / Length * DT
  • Represents: A correction term that accounts for the change in cross-sectional area over time. This term is important when the flow or water level changes rapidly.
  • Units:
    • K: A conversion factor with units of feet/second
    • Q: feet^3/second
    • Area: feet^2
    • Length: feet
    • DT: seconds
    • Result: Dimensionless (all units cancel out). This term also acts as a scaling factor in the denominator of the main equation.

Figures 1, 2, and 3: Visualizing the St. Venant Terms

The figures you provided are excellent for visualizing the contributions of each term:

• Figure 1: Shows the absolute values of each term over time. This helps to see which terms are dominant at different points in the simulation.
• Figure 2: Shows the relative magnitudes of the terms, making it easier to compare their contributions.
• Figure 3: Presents the relative magnitudes as an area chart, normalized to 100%. This clearly illustrates the percentage contribution of each term to the overall flow calculation at each time step.

Key Observations from the Figures:

• DQ1 and DQ2 Balance: As you noted, DQ1 (friction) and DQ2 (pressure) often balance each other, especially during steady or gradually varied flow.
• DQ3 and DQ4 Dominate in Unsteady Flow: During periods of rapid change (backwater, flow reversal, rapid filling or emptying), DQ3 (convective acceleration) and DQ4 (local acceleration) become more significant, reflecting the dynamic nature of the flow.

 Summary

A comprehensive and accurate description of the five St. Venant flow terms in SWMM 5 and InfoSWMM. Understanding these terms, their units, and their contributions is essential for interpreting dynamic wave routing results and for troubleshooting any issues that may arise during a simulation. The figures visualize the relative importance of each term under different flow conditions. Remember that these five terms, along with the iterative solution method, are what give SWMM 5 the power to accurately simulate complex, unsteady flow in a variety of hydraulic systems.

Figure 1: The Five St. Venant Components over time.

Figure 2: The relative magnitude of the St Venant terms over time for the same link as in Figure 1.

Figure 3: Normalized Area Chart of St. Venant Terms

• Purpose: To show the relative contribution of each of the five St. Venant terms (DQ1, DQ2, DQ3, DQ4, and DQ5) to the overall flow calculation at each time step.
• Normalization: The values of each term are scaled so that they add up to 100%. This makes it easy to see the percentage that each term contributes to the new flow calculation in each iteration.
• Area Chart: The use of an area chart visually represents the changing dominance of each term over time. The area occupied by each term's "band" in the chart is proportional to its percentage contribution.

Interpreting the Figure:

• DQ1 (Friction) and DQ2 (Pressure) Balance: Under normal or gradually varied flow conditions, you'll typically observe that DQ1 and DQ2 are the dominant terms and that they often roughly balance each other out. This is because:

  • DQ1 (Friction): Represents the force resisting the flow due to friction along the conduit.
  • DQ2 (Pressure): Represents the force driving the flow due to the pressure gradient (difference in water surface elevation).
  • In a balanced state, the force of gravity (represented in the pressure term) is counteracted by the frictional resistance.

• DQ3 (Convective Acceleration) and DQ4 (Local Acceleration) Dominate in Dynamic Conditions:

  • Backwater: When a downstream obstruction or high water level creates backwater, the flow slows down and may even reverse. In these situations, the convective and local acceleration terms (DQ3 and DQ4) become more significant as they account for the changes in momentum caused by the changing velocity and flow area.
  • Reverse Flow: Similar to backwater, when the flow direction reverses, the velocity changes rapidly, and DQ3 and DQ4 will play a larger role in the flow calculation.
  • Rapid Changes in Flow: Any situation with rapidly changing flow conditions (e.g., sudden gate openings or closures, rapid filling or emptying of storage) will also see increased contributions from DQ3 and DQ4.

• DQ5 (Area Correction): This term is often smaller than the others but becomes important when the flow or water level changes very rapidly. It helps to maintain numerical stability during these dynamic conditions.

Value of Figure 3:

• Understanding Model Behavior: This type of visualization is invaluable for gaining insight into how the model is behaving and why the flow is changing in a particular way. It helps you understand which physical forces are dominant at different times in the simulation.
• Troubleshooting: If you encounter unexpected results or instability in your model, examining the relative contributions of the St. Venant terms can help pinpoint the cause. For example, if DQ3 and DQ4 are consistently very large, it might indicate a problem with the time step or other model parameters.
• Model Validation: By comparing the model's behavior to your understanding of the physical system, you can assess whether the model is representing the real-world processes realistically.

Example Scenario from Figure 3:

Let's imagine a hypothetical scenario based on a potential Figure 3:

1. Initial Period: DQ1 and DQ2 are dominant and roughly equal, indicating steady or gradually varied flow.
2. Sudden Increase in Inflow: A storm event causes a rapid increase in inflow. You might see DQ3 and DQ4 spike as the flow accelerates. DQ5 might also increase to maintain stability.
3. Backwater Develops: As the increased flow reaches a downstream constriction, backwater develops. DQ3 and DQ4 might become even more prominent, potentially with negative values if flow reversal occurs.
4. Flow Recedes: As the storm subsides, the inflow decreases. DQ1 and DQ2 might gradually return to their dominant, balanced state.